{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/102425"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/102425","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Lipschitz and Holder mappings into jet space Carnot groups","abstract":"\"For $k,n\\ge 1$, the jet space $J^k(\\R^n)$ is the set of $k^{th}$-order Taylor polynomials of functions in $C^k(\\R^n)$. Warhurst constructs a Carnot group structure on $J^k(\\R^n)$ such that the jets of functions in $C^{k+1}(\\R^n)$ are horizontal. Like in all Carnot groups, one can define a Carnot-Carath\\'eodory metric on $J^k(\\R^n)$ by minimizing lengths of horizontal paths. Unfortunately, exact forms or even the regularities of geodesics connecting generic pairs of points are not known for $J^k(\\R^n)$. After describing the Carnot group structure of $J^k(\\R^n)$, we will prove that there exists a biLipschitz embedding of $\\mathbb{S}^n$ into $J^k(\\R^n)$ that does not admit a Lipschitz extension to $\\mathbb{B}^{n+1}$. This strengthens a result of Rigot and Wenger \\cite{RW:LNE} and generalizes a result for $\\mathbb{H}^n$ of Dejarnette, Haj{\\l}asz, Lukyanenko, and Tyson. We will then consider a problem related to Gromov's conjecture on the H\\\"\"older equivalence of Carnot groups. We will prove that for all $m\\ge 2$ and $\\epsilon>0$, there does not exist an injective, locally $(\\frac{1}{2}+\\epsilon)$-H\\\"\"older mapping $f:\\R^m\\to J^k(\\R)$ that is locally Lipschitz as a mapping into $\\R^{k+2}$. This builds on a result of Balogh, Haj{\\l}asz, and Wildrick for $\\mathbb{H}^n$. We will conclude by proposing analogues of horizontal and vertical projections for $J^k(\\R)$. We prove Marstrand-type results for these mappings. This continues efforts of Balogh, Durand-Cartagena, F\\\"\"assler, Mattila, and Tyson over the past decade to prove Marstrand-type theorems in a sub-Riemannian setting. We will study the metric structure of $J^k(\\R^n)$, focusing primarily on the model filiform jet spaces $J^k(\\R)$.\"","abstract_html":"&quot;For $k,n\\ge 1$, the jet space <span class=\"etd-inline-math\">J<sup>k</sup>(\\R<sup>n</sup>)</span> is the set of <span class=\"etd-inline-math\">k<sup>th</sup></span>-order Taylor polynomials of functions in <span class=\"etd-inline-math\">C<sup>k</sup>(\\R<sup>n</sup>)</span>. Warhurst constructs a Carnot group structure on <span class=\"etd-inline-math\">J<sup>k</sup>(\\R<sup>n</sup>)</span> such that the jets of functions in <span class=\"etd-inline-math\">C<sup>k+1</sup>(\\R<sup>n</sup>)</span> are horizontal. Like in all Carnot groups, one can define a Carnot-Carath\\&#x27;eodory metric on <span class=\"etd-inline-math\">J<sup>k</sup>(\\R<sup>n</sup>)</span> by minimizing lengths of horizontal paths. Unfortunately, exact forms or even the regularities of geodesics connecting generic pairs of points are not known for <span class=\"etd-inline-math\">J<sup>k</sup>(\\R<sup>n</sup>)</span>. After describing the Carnot group structure of <span class=\"etd-inline-math\">J<sup>k</sup>(\\R<sup>n</sup>)</span>, we will prove that there exists a biLipschitz embedding of <span class=\"etd-inline-math\">\\mathbb{S}<sup>n</sup></span> into <span class=\"etd-inline-math\">J<sup>k</sup>(\\R<sup>n</sup>)</span> that does not admit a Lipschitz extension to <span class=\"etd-inline-math\">\\mathbb{B}<sup>n+1</sup></span>. This strengthens a result of Rigot and Wenger \\cite{RW:LNE} and generalizes a result for <span class=\"etd-inline-math\">\\mathbb{H}<sup>n</sup></span> of Dejarnette, Haj{\\l}asz, Lukyanenko, and Tyson. We will then consider a problem related to Gromov&#x27;s conjecture on the H\\&quot;&quot;older equivalence of Carnot groups. We will prove that for all $m\\ge 2$ and <span class=\"etd-inline-math\">&epsilon;&gt;0</span>, there does not exist an injective, locally <span class=\"etd-inline-math\">(\\frac{1}{2}+&epsilon;)</span>-H\\&quot;&quot;older mapping <span class=\"etd-inline-math\">f:\\R<sup>m</sup>\\to J<sup>k</sup>(\\R)</span> that is locally Lipschitz as a mapping into <span class=\"etd-inline-math\">\\R<sup>k+2</sup></span>. This builds on a result of Balogh, Haj{\\l}asz, and Wildrick for <span class=\"etd-inline-math\">\\mathbb{H}<sup>n</sup></span>. We will conclude by proposing analogues of horizontal and vertical projections for <span class=\"etd-inline-math\">J<sup>k</sup>(\\R)</span>. We prove Marstrand-type results for these mappings. This continues efforts of Balogh, Durand-Cartagena, F\\&quot;&quot;assler, Mattila, and Tyson over the past decade to prove Marstrand-type theorems in a sub-Riemannian setting. We will study the metric structure of <span class=\"etd-inline-math\">J<sup>k</sup>(\\R<sup>n</sup>)</span>, focusing primarily on the model filiform jet spaces <span class=\"etd-inline-math\">J<sup>k</sup>(\\R)</span>.&quot;","abstract_has_math":true,"creators":["Jung, Derek"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Tyson, Jeremy","Wu, Jang-Mei","Fernandes, Rui","Kaufman, Robert"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2019,"date_issued":"2019-02-06T19:32:49Z","date_published":"2019-02-06T19:32:49Z","updated_at":"2026-07-22T22:24:40Z","subjects":["Sub-Riemannnian Geometry","Jet spaces","Carnot groups","Geometric Analysis"],"languages":["en"],"rights":["Copyright 2018 Derek Jung"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/102425","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Tyson, Jeremy","Wu, Jang-Mei","Fernandes, Rui","Kaufman, Robert"]},{"key":"dc:creator","label":"Author","values":["Jung, Derek"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2019-02-06T19:32:49Z","2018-11-08","2018-12"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Sub-Riemannnian Geometry","Jet spaces","Carnot groups","Geometric Analysis"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2018 Derek Jung"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/102425"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["\"For $k,n\\ge 1$, the jet space $J^k(\\R^n)$ is the set of $k^{th}$-order Taylor polynomials of functions in $C^k(\\R^n)$. Warhurst constructs a Carnot group structure on $J^k(\\R^n)$ such that the jets of functions in $C^{k+1}(\\R^n)$ are horizontal. Like in all Carnot groups, one can define a Carnot-Carath\\'eodory metric on $J^k(\\R^n)$ by minimizing lengths of horizontal paths. Unfortunately, exact forms or even the regularities of geodesics connecting generic pairs of points are not known for $J^k(\\R^n)$. After describing the Carnot group structure of $J^k(\\R^n)$, we will prove that there exists a biLipschitz embedding of $\\mathbb{S}^n$ into $J^k(\\R^n)$ that does not admit a Lipschitz extension to $\\mathbb{B}^{n+1}$. This strengthens a result of Rigot and Wenger \\cite{RW:LNE} and generalizes a result for $\\mathbb{H}^n$ of Dejarnette, Haj{\\l}asz, Lukyanenko, and Tyson. We will then consider a problem related to Gromov's conjecture on the H\\\"\"older equivalence of Carnot groups. We will prove that for all $m\\ge 2$ and $\\epsilon>0$, there does not exist an injective, locally $(\\frac{1}{2}+\\epsilon)$-H\\\"\"older mapping $f:\\R^m\\to J^k(\\R)$ that is locally Lipschitz as a mapping into $\\R^{k+2}$. This builds on a result of Balogh, Haj{\\l}asz, and Wildrick for $\\mathbb{H}^n$. We will conclude by proposing analogues of horizontal and vertical projections for $J^k(\\R)$. We prove Marstrand-type results for these mappings. This continues efforts of Balogh, Durand-Cartagena, F\\\"\"assler, Mattila, and Tyson over the past decade to prove Marstrand-type theorems in a sub-Riemannian setting. We will study the metric structure of $J^k(\\R^n)$, focusing primarily on the model filiform jet spaces $J^k(\\R)$.\"","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2019-02-05 without embargo terms","The student, Derek Jung, accepted the attached license on 2018-11-06 at 07:31.","The student, Derek Jung, submitted this Dissertation for approval on 2018-11-06 at 07:40.","This Dissertation was approved for publication on 2018-11-08 at 09:07.","DSpace SAF Submission Ingestion Package generated from Vireo submission #13061 on 2019-02-05 at 11:08:54","Made available in DSpace on 2019-02-06T19:32:49Z (GMT). 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Warhurst constructs a Carnot group structure on $J^k(\\R^n)$ such that the jets of functions in $C^{k+1}(\\R^n)$ are horizontal. Like in all Carnot groups, one can define a Carnot-Carath\\'eodory metric on $J^k(\\R^n)$ by minimizing lengths of horizontal paths. Unfortunately, exact forms or even the regularities of geodesics connecting generic pairs of points are not known for $J^k(\\R^n)$. After describing the Carnot group structure of $J^k(\\R^n)$, we will prove that there exists a biLipschitz embedding of $\\mathbb{S}^n$ into $J^k(\\R^n)$ that does not admit a Lipschitz extension to $\\mathbb{B}^{n+1}$. This strengthens a result of Rigot and Wenger \\cite{RW:LNE} and generalizes a result for $\\mathbb{H}^n$ of Dejarnette, Haj{\\l}asz, Lukyanenko, and Tyson. We will then consider a problem related to Gromov's conjecture on the H\\\"\"older equivalence of Carnot groups. We will prove that for all $m\\ge 2$ and $\\epsilon>0$, there does not exist an injective, locally $(\\frac{1}{2}+\\epsilon)$-H\\\"\"older mapping $f:\\R^m\\to J^k(\\R)$ that is locally Lipschitz as a mapping into $\\R^{k+2}$. This builds on a result of Balogh, Haj{\\l}asz, and Wildrick for $\\mathbb{H}^n$. We will conclude by proposing analogues of horizontal and vertical projections for $J^k(\\R)$. We prove Marstrand-type results for these mappings. This continues efforts of Balogh, Durand-Cartagena, F\\\"\"assler, Mattila, and Tyson over the past decade to prove Marstrand-type theorems in a sub-Riemannian setting. We will study the metric structure of $J^k(\\R^n)$, focusing primarily on the model filiform jet spaces $J^k(\\R)$.\"","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2019-02-05 without embargo terms","The student, Derek Jung, accepted the attached license on 2018-11-06 at 07:31.","The student, Derek Jung, submitted this Dissertation for approval on 2018-11-06 at 07:40.","This Dissertation was approved for publication on 2018-11-08 at 09:07.","DSpace SAF Submission Ingestion Package generated from Vireo submission #13061 on 2019-02-05 at 11:08:54","Made available in DSpace on 2019-02-06T19:32:49Z (GMT). No. of bitstreams: 6 JUNG-DISSERTATION-2018.pdf: 545800 bytes, checksum: 7552089aec29e8d004b6453984c42e8b (MD5) ThesisJung.bib: 41450 bytes, checksum: fc920505d125290eaa05a78b45121911 (MD5) Thesis_jung_Nov41209.bbl: 10027 bytes, checksum: 151cdbe145dd2a020e4a140a8d01900b (MD5) Thesis_jung_Nov41209.tex: 297189 bytes, checksum: d5eb59dcd20df1c79139ec45ef0a1d6e (MD5) LICENSE.txt: 4207 bytes, checksum: 0c96b859403c9fbce7a0c55577afec26 (MD5) PROQUEST_LICENSE.txt: 4553 bytes, checksum: cb50808d7910c57a6b5d4ca2cb4fcb4d (MD5) Previous issue date: 2018-11-08"],"dc:format":["application/pdf"],"dc:identifier":["http://hdl.handle.net/2142/102425"],"dc:language":["en"],"dc:rights":["Copyright 2018 Derek Jung"],"dc:subject":["Sub-Riemannnian Geometry","Jet spaces","Carnot groups","Geometric Analysis"],"dc:title":["Lipschitz and Holder mappings into jet space Carnot groups"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:24:40Z"}